An Upper Bound for Spherical Caps

We prove an useful upper bound for the measure of spherical caps. Consider the uniformly distributed measure σn−1 on the Euclidean unit sphere Sn−1 ⊂ R. On the sphere, as among only a handful other spaces, the isoperimetric problem is completely solved. This goes back to Lévy [Lé] and Schmidt [Sch] and states that caps have the minimal measure of a boundary among all sets with a fixed mass. For ε ∈ [0, 1) and θ ∈ Sn−1 the cap C(ε, θ), or shortly C( ), is a set of points x ∈ Sn−1 for which x · θ ≥ ε, where · stands for the standard scalar product in R. See figure 1. Figure 1: A cap C(ε, θ). A few striking properties of the high-dimensional sphere are presented in [Ba, Lecture 1, 8]. In such considerations, we often need a good estimation of the measure of a cap. Following the method used in [Ba, Lemma 2.2], we extend its proof to the skipped case of large ε and get in an elementary way the desired bound. Theorem. For any ε ∈ [0, 1) σn−1 (C(ε)) ≤ e−nε 2/2. Figure 2: Small ε. Proof. In the case of small ε, for convenience, we repeat a beautiful argument used by Ball. Namely, for ε ∈ [0, 1/ √ 2] we have (see Figure 2) σn−1 (C(ε)) = voln (Cone ∩B(0, 1)) voln (Bn(0, 1)) ≤ voln ( B(P, √ 1− ε2) ) voln (Bn(0, 1)) = √ 1− ε2 n ≤ e−nε 2/2. For ε ∈ [1/ √ 2, 1), it is enough to consider a different auxiliary ball which includes the set Cone ∩B(0, 1), see Figure 3. We obtain σn−1 (C(ε)) ≤ voln (B (Q, r))) voln (Bn(0, 1)) = r = ( 1 2ε )n ≤ e−nε 2/2, where the last inequality follows from the estimate Figure 3: Large ε. By the congruence 1/2 r = ε 1 . ∗Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland. [email protected]